Let f: R → R be defined as f(x)= [[e^x], x<0; a e^x+[x-1], 0 ≤ x<1; b+[ sin (π x)], 1 ≤ x<2; [e^-x]-c, x ≥ 2 where a, b, c ∈ R and [t] denotes…

Let $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as $$f(x)=\left[\begin{array}{ll} {\left[e^x\right],} & x<0 \\ a e^x+[x-1], & 0 \leq x<1 \\ b+[\sin (\pi x)], & 1 \leq x<2 \\ {\left[e^{-x}\right]-c,} & x \geq 2 \end{array}\right. $$ where $\displaystyle a, b, c \in \mathbb{R}$ and $\displaystyle [t]$ denotes greatest integer less than or equal to $\displaystyle t$. Then, which of the following statements is true ?